Rotating the Hopf Fibration

The Hopf fibration describes a relationship between the one-dimensional sphere (a circle), the two-dimensional sphere (an ordinary sphere), and the three-dimensional sphere (a hypersphere in 4D space) as a fibration with as the fiber, as the base space, and as the total space. This mapping has the property that when viewed locally, is indistinguishable from the Cartesian product . However, this is not true globally, since a fibration has a "twist" that distinguishes it from a regular product space.

This Demonstration allows you to manipulate a set of points in (shown in the bottom-left corner) and view the corresponding circles in with stereographic projection, revealing much of the interesting structure induced by the Hopf map.